"21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.
aₙ₊₁ = 3aₙ-12; a₁ = 10

Verified step by step guidance

1. Identify the given recurrence relation: \(a_{n+1} = 3a_n - 12\) with the initial term \(a_1 = 10\).
2. Calculate the second term \(a_2\) by substituting \(n=1\) into the recurrence relation: \(a_2 = 3a_1 - 12\).
3. Calculate the third term \(a_3\) by substituting \(n=2\): \(a_3 = 3a_2 - 12\).
4. Calculate the fourth term \(a_4\) by substituting \(n=3\): \(a_4 = 3a_3 - 12\).
5. List the first four terms of the sequence as \(a_1\), \(a_2\), \(a_3\), and \(a_4\) after computing each step.

Key Concepts

Recurrence Relations

A recurrence relation defines each term of a sequence using previous terms. It provides a formula to generate terms step-by-step, starting from initial values. Understanding how to apply the relation repeatedly is essential to find subsequent terms.

Initial Conditions

Initial conditions specify the starting values of a sequence, such as a₁ = 10. These values are necessary to begin the process of generating terms using the recurrence relation, as they anchor the sequence and allow computation of all following terms.

Iterative Computation of Terms

To find terms of a sequence defined by a recurrence relation, you substitute known terms into the relation repeatedly. This iterative process involves calculating each term step-by-step, using the previous term(s) and the given formula.